For almost as long as there have been consoles, manufacturers have created additional hardware to expand their capabilities.

Which do you think is the best such add-on/peripheral, and what makes your choice the best?

Just from the concept (my cuz only let me use it a few times), Game Boy Player is perhaps the best.
It gives you access to 99% of the GB, GBC, and GBA libraries.
Super Game Boy also deserves some love for doing it first, but you obviously can't use any of the GBA library and none of the Game Boy Color clear cartridges that don't work on Game Boy.
And that's it.

I suppose PSVR counts so that but if it does not, this thing brought be great joy specifically with Time crisis.

The Link Cable hands down. Nowadays it may not seem like a big deal... But back then, being able to connect your Gameboy with your friend's? Trading Pokémon? Fighting each other? Going "back in time" and trade with RBY from GSC? It was like goddamn magic! It not only enhanced the games immensely, they magnified the social aspect of Pokémon to insane levels.

PC Engine CD/TurboGrax-1 CD. The very first system with the ability to play games on CD and guess what. It was amazing. A lot of the system's best games are on it. From Ys. To Cosmic Fantasy. Castlevania Rondo of Blood. A ton of amazing shmups. With CD-quality music. Voice acting (hit and miss but was there) The music esp was fantastic. That Ys intro still gives me chills. SEGA CD had some great stuff and gets a worse rep than it deserves but SCD did not hold a candle to how amazing the PCECD add-on was.

Hands down the psvr2 for the ps5. What an improvement to psvr1. Incredible immersion. GT7, RE4, RE8, kayak, tetris effect, moss, really good stuff and it transforms to normal ps5 experience into something completely different.

Well, by sales alone, Wii Balance Board (42 millions). As someone who still has it, it was always fun accessory, both for fitness and actual games.

By both sales (35 millions) and actual transformative power of making standard home console into something quite different, I'd go with Kinect.

By transformative power alone, I'd go with something VR, cause that opens up pretty much everything.

Does something like the WaveBird count?

Might be a strange take but the Circle Pad Pro for 3DS might take the title for me. Simply because many games with 3D environments felt unplayable without that extra control stick. The device was ugly and not very comfortable to use. Mostly a hardware to fix a glaring flaw in the original 3DS design.

I still use mine, 16-17 years later. It works just as great as a coffee for getting me going in the morning. Remarkably durable. Although, I hope there’s a more proper update than Ring Fit 2 on Switch 2.

The GameBoy Player, both in it's Super Nintendo an in it's GameCube incarnation. In fact, my GameCube is mainly active just to play those Games on the big screen these days.